Corollary 2
Tag 04BL, permanent. Cite it as [Bourbaki, Tag 04BL].
Commutative Algebra, chapter II, § 5. Finitely generated projective modules. Invertible fractional ideals
Every finitely presented flat module is projective.
If P is a finitely presented flat A-module and m a maximal ideal of A, the -module P, is flat (§ 3, no. 4, Proposition 13) and finitely presented (§ 2, no. 4) and hence free (§ 3, no. 2, Corollary 2 to Proposition 5), Condition (b) of Theorem 1 therefore holds.
Remarks (1) There exist finitely generated flat modules which are not projective (Exercise 7). (2) Corollary 2 to Theorem 1 extends to modules over non-commutative rings (Chapter I, §2, Exercise 15).
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.